Nature's Numbers: The Unreal Reality of Mathematics (Vocabulary Flashcards)

Vocabulary flashcards covering key terms and concepts from Nature's Numbers: The Unreal Reality of Mathematics by Ian Stewart.

Fractals

Geometric shapes that repeat their structure at ever-finer scales; exhibit self-similarity and scale invariance.

Chaos

Apparent randomness that arises from deterministic rules; chaotic systems are deterministic but unpredictable due to sensitivity to initial conditions.

Fractal geometry

A branch of mathematics studying irregular shapes that display self-similarity and roughness across scales.

Self-similarity

A property where a part of an object resembles the whole object, typical of fractals and many natural patterns.

Phase space

An abstract space representing all possible states of a dynamical system; coordinates are the system’s variables.

Attractor

A set or shape in phase space toward which a system tends to evolve; examples include fixed points, limit cycles, and strange attractors.

Limit cycle

A closed trajectory in phase space representing a periodic oscillation or rhythm.

Strange attractor

A chaotic attractor with a complex, fractal structure; sensitive dependence on initial conditions.

Phase portrait

A geometric depiction of trajectories in phase space showing how system states evolve.

Feigenbaum constant delta

A universal constant (~4.669) governing period-doubling cascades leading to chaos.

Period-doubling cascade

Sequence of bifurcations where the period of oscillation doubles, often leading to chaos.

Hopf bifurcation

A temporal symmetry-breaking transition where a steady state becomes periodic, creating oscillations.

Central pattern generator (CPG)

Neural circuits that produce rhythmic patterns (e.g., gaits) with little or no sensory input.

Gait

Pattern of limb movement during locomotion (walk, trot, pace, bound, canter, gallop, pronk).

Symmetry

A transformation that leaves an object invariant (e.g., reflections, rotations, translations).

Broken symmetry

When a symmetric system ends up in a less symmetric state; often a fundamental mechanism in pattern formation.

Mirror symmetry

Symmetry under reflection across a mirror line or plane.

Time-translation symmetry

Invariance of a system’s behavior under shifting the time origin; broken in oscillations like Hopf bifurcations.

Inverse-square law

A force law where force ∝ 1/r^2 (e.g., gravity between masses).

Newton’s laws of motion

F = m a; force equals mass times acceleration; describes how motion responds to forces.

Calculus

Branch of mathematics dealing with rates of change (differentiation) and accumulation (integration).

Differentiation

Finding the rate at which a quantity changes; the derivative.

Integration

Finding the total or accumulated quantity; the inverse process of differentiation.

Real numbers

Numbers on a continuous line including rationals and irrationals; decimals go on forever.

Rational numbers

Numbers that can be expressed as a ratio of integers (fractions).

Irrational numbers

Numbers not expressible as fractions (e.g., sqrt(2)); non-terminating, non-repeating decimals.

Complex numbers

Numbers of the form a + bi; include imaginary unit i where i^2 = -1.

Imaginary numbers

Part of complex numbers; involve the square root of negative numbers.

Functions

A rule that assigns to each input a unique output; also called a transformation.

Thingification (reification)

Treating processes as if they were objects or things; turning actions into mathematical entities.

Data structures

Organized collections of data (e.g., lists, arrays) used in computing.

Golden ratio (phi)

Approximately 1.618; the limit of the ratio of consecutive Fibonacci numbers; appears in growth patterns.

Golden angle

Approximately 137.5 degrees; divergence angle between successive plant primordia yielding efficient packing.

Fibonacci numbers

Sequence where each number is the sum of the two preceding ones (3,5,8,13,21,34,55,…).

Kepler’s ellipses

Planets move in ellipses; a key step toward Newtonian gravity.

Ellipse

An oval; the generalization of a circle; a basic conic section in orbital mechanics.

Resonance

A relationship between periodic motions with commensurate periods; cycles lock into simple ratios.

Three-body problem

Motion of three masses under gravity; not solvable in closed form; often chaotic.

Two-body problem

Motion of two bodies under gravity; integrable with exact solutions (ellipses).

Belousov–Zhabotinsky reaction (B-Z reaction)

A family of oscillating chemical reactions that form temporal or spatial patterns (e.g., target patterns, spirals).

Phase space attractors in biology

Geometric objects in phase space that describe long-term behavior of biological systems (e.g., gait patterns, population cycles).

Morphomatics

A proposed field aiming to unify mathematics of form, patterns, and development across disciplines.